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Economic Capital and the Aggregation of Risks Using Copulas

Economic Capital and the Aggregation of Risks Using Copulas Dr. Emiliano A. Valdez and Andrew Tang. Motivation and aims Technical background - copulas Numerical simulation Results of simulation Key findings and conclusions. Overview. Capital. Buffer

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Economic Capital and the Aggregation of Risks Using Copulas

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  1. Economic Capital and the Aggregation of Risks Using Copulas Dr. Emiliano A. Valdez and Andrew Tang

  2. Motivation and aims Technical background - copulas Numerical simulation Results of simulation Key findings and conclusions Overview

  3. Capital • Buffer A rainy day fund, so when bad things happen, there is money to cover it Quoted from the IAA Solvency Working Party (2004) – “A Global Framework for Solvency Assessment” • Solvency and financial strength indicator • Economic capital - worst tolerable value of the risk portfolio

  4. Multi-Line Insurers • Increasingly prominent • Diverse range insurance products • Aggregate loss, Z Where Xi represents the loss variable from line i. • Xis are dependent

  5. Multi-Line Insurers • Dependencies between Xis ignored • E.g., APRA Prescribed Method • Dependencies modelled using linear correlations • Inadequate • Non-linear dependence • Tail dependence

  6. Multi-Line Insurers • Capital risk measures • Capital requirements • Value-at-Risk (VaR) – quantile risk measure • Tail conditional expectation (TCE)

  7. Multi-Line Insurers • Diversification benefit • q = 97.5% and 99.5%

  8. Aims • Study the capital requirements (CRs) under different copula aggregation models • Study the diversification benefits (DBs) under different copula aggregation models • Compare the CRs from copula models to the Prescribed Method (PM) used by APRA

  9. Copulas • Individual line losses - X1, X2, …, Xn • Joint distribution is F(x1,x2,…,xn) • Marginal distributions are F1(x1), F2(x2), …, Fn(xn) • A copula, C, is a function that links, or couples the marginals to the joint distribution • Sklar (1959)

  10. Copulas • Copulas of extreme dependence • Independence copula • Archimedean copulas • Gumbel-Hougaard copula • Frank copula • Cook-Johnson copula

  11. Copulas • Elliptical copulas / variants of the student-t copula • Gaussian “Normal” copula (infinite df) • Student-t copula (3 & 10 df) • Cauchy copula (1 df) Where Tv(.) and tv(.) denote the multivariate and univariate Student-t distribution with v degrees of freedom respectively.

  12. Copulas • Tail dependence (Student-t copulas) where t* denotes the survivorship function of the Student-t distribution with n degrees of freedom.

  13. Numerical Simulation • 1 year prospective gross loss ratios for each line of business • Industry data between 1992 and 2002 • Semi-annual • SAS/IML (Interactive Matrix Language)

  14. Numerical Simulation • Five lines of business • Motor: domestic & commercial • Household: buildings & contents • Fire & ISR • Liability: public, product, WC & PI • CTP

  15. Numerical Simulation • Correlation matrix input

  16. Numerical Simulation • Marginal distribution input

  17. Results of Simulation • Normal copula

  18. Results of Simulation • Student-t (3 df) copula

  19. Results of Simulation • Student-t (10 df) copula

  20. Results of Simulation • Cauchy copula

  21. Results of Simulation • Independence copula

  22. Results of Simulation • Aggregated loss, Z, under each copula

  23. Results of Simulation • Capital requirements (CRs) Note: risk measures 1 – 4 are VaR(97.5%), VaR(99.5%),TCE(97.5%) and TCE(99.5%) respectively.

  24. Results of Simulation • Diversification benefits (DBs) Note: risk measures 1 – 4 are VaR(97.5%), VaR(99.5%),TCE(97.5%) and TCE(99.5%) respectively.

  25. Results of Simulation • Comparison with Prescribed Method (PM) – industry portfolio

  26. Results of Simulation • Comparison with Prescribed Method (PM) – short tail portfolio

  27. Results of Simulation • Comparison with Prescribed Method (PM) – long tail portfolio

  28. Key Findings • Choice of copula matters dramatically for both CRs and DBs • More tail dependent  higher CR • More tail dependent  higher DB • Need to select the correct copula for the insurer’s specific dependence structure • CR and DB shares a positive relationship • PM is not a “one size fits all” solution • Mis-estimations of the true capital requirement

  29. Limitations • Simplifying assumptions • Underwriting risk only • Ignores impact of reinsurance • Ignores impact of investment • Results do not quantify the amount of capital required • Comparison between copulas • Not comparable with results of other studies

  30. Further Research • Other copulas • Isaacs (2003) used the Gumbel • Other types of risk dependencies • E.g., between investment and operational risks • Relax some assumptions • Include reinsurance • Factor in expenses • Factor in investments

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